Advanced Calculus of Several Variables (1973)
Part V. Line and Surface Integrals; Differential Forms and Stokes' Theorem
Chapter 6. STOKES' THEOREM
Recall that Green's theorem asserts that, if ω is a differential 1-form on the cellulated region , then
Stokes' theorem is a far-reaching generalization which says the same thing for a differential (k − 1)-form α which is defined on a neighborhood of the cellulated k-dimensional region , where M is a smooth k-manifold in n. That is,
Of course we must say what is meant by a cellulated region in a smooth manifold, and also what the above integrals mean, since we thus far have defined integrals of differential forms only on surface patches, cells, and manifolds. Roughly speaking, a cellulation of R will be a collection {A1, . . . , Ar} of nonoverlapping k-cells which fit together nicely (like the 2-cells of a cellulation of a planar region), and the integral of a k-form ω on R will be defined by
Green‘s theorem is simply the case n = k = 2 of Stokes’ theorem, while the fundamental theorem of (single-variable) calculus is the case n = k = 1 (in the sense of the discussion at the beginning of Section 2 ). We will see that other cases as well have important physical and geometric applications. Therefore Stokes' theorem may justly be called the “fundamental theorem of multivariable calculus.”
The proof in this section of Stokes’ theorem will follow the same pattern as the proof in Section 2 of Green’s theorem. That is, we will first prove Stokes’ theorem for the unit cube . This simplest case will next be used to establish Stokes’ theorem for a k-cell on a smooth k-manifold in n, using the formal properties of the differential and pullback operations. Finally the general result, for an appropriate region R in a smooth k-manifold, will be obtained by application of Stokes' theorem to the cells of a cellulation of R.
So we start with the unit cube
in Rk. Its boundary ∂Ik is the set of all those points for which one of the k coordinates is either 0 or 1; that is, ∂Ik is the union of the 2k different (k − 1)-dimensional faces of Ik. These faces are the sets defined by
for each i = 1, . . . , k and = 0 or = 1 (see Fig. 5.32) . The (i, )the face of Ik is the image of the unit cube under the mapping li, : Ik−1 → k defined by
The mapping li, serves as an orientation for the face . The orientations which the faces (edges) of I2 receive from these mappings are indicated by the
Figure 5.32
arrows in Fig. 5.32. We see that the positive (counterclockwise) orientation of ∂I2 is given by
If ω is a continuous 1-form, it follows that the integral of ω over the oriented piecewise closed curve ∂I2 is given by
The integral of a (k − 1)-form α over ∂Ik is defined by analogy. That is, we define
As in Section 5 , the integral over Ik of the k-form f dx1 · · · dxk is defined by
We are now ready to state and prove Stokes’ theorem for the unit cube. Its proof, like that of Green’s theorem for the unit square (Lemma 2.1) , will be an explicit computation.
Theorem 6.1 If α is a differential (k − 1)-form that is defined on an open set containing the unit cube , then
PROOF Let α be given by
where a1, . . . , ak are real-valued functions on a neighborhood of Ik. Then
since
We compute both sides of Eq. (2) . First
where
because
To compute the left-hand side of (2) , we first apply Fubini's theorem and the (ordinary) fundamental theorem of calculus to obtain
using (3) in the last step. Therefore
as desired.
Notice that the above proof is a direct generalization of the computation in the proof of Green‘s theorem for I2, using only the relevant definitions, Fubini’s theorem, and the single-variable fundamental theorem of calculus.
The second step in our program is the proof of Stokes's theorem for an oriented k-cell in a smooth k-manifold in n. Recall that a (smooth) k-cell in the smooth k-manifold is the image A of a mapping φ : Ik → M which extends to a coordinate patch for M (defined on some open set in k which contains Ik). If M is oriented, then we will say that the k-cell A is oriented (positively with respect to the orientation of M) if this coordinate patch is orientation-preserving. To emphasize the importance of the orientation for our purpose here, let us make the following definition. An oriented k-cell in the oriented smooth k-manifold is a pair (A, φ), where φ : Ik → M is a mapping which extends to an orientation-preserving coordinate patch, and A = φ(Ik) (see Fig. 5.33) .
The (i, )th face Ai, of A is the image under φ of the (i, )th face of . Thus Ai, is the image of Ik−1 under the mapping
If α is a (k − 1)-form which is defined on A, we define the integral of α over the oriented boundary ∂A by analogy with Eq. (1) ,
Also, if β is a k-form defined on A, we write
With this notation, Stokes' theorem for an oriented k-cell takes the following form.
Figure 5.33
Theorem 6.2 Let (A, φ) be an oriented k-cell in an oriented k-manifold in n, and let α be a differential (k − 1)-form that is defined on an open subset of n which contains A. Then
PROOF The proof is a computation which is a direct generalization of the proof of Green‘s theorem for oriented 2-cells (Lemma 2.3) . Applying Stokes’ theorem for Ik, and the formal properties of the pullback and differential operations, given in Theorems 5.4 and 5.5 , respectively, we obtain
as desired.
Our final step will be to extend Stokes' theorem to regions that can be obtained by appropriately piecing together oriented k-cells. Let R be a compact region (the closure of an open subset) in the smooth k-manifold . By an oriented cellulation of R is meant a collection = {A1, . . . , Ap} of oriented k-cells (Fig. 5.34) on M satisfying the following conditions:
(a)
(b)For each r and s, the intersection is either empty or is the union of one or more common faces of Ar and As.
Figure 5.34
(c)If B is a (k − 1)-dimensional face of Ar, then either Ar is the only k-cell of having B as a face, or there is exactly one other k-cell As also having B as a face. In the former case B is called a boundary face; in the latter B is an interior face.
(d)If B is an interior face, with Ar and As the two k-cells of having B as a face, then Ar and As induce opposite orientations on B.
(e)The boundary ∂R of R (as a subset of M) is the union of all the boundary faces of k-cells of .
The pair (R, ) is called an oriented cellulated region in M.
Although an oriented cellulation is conceptually simple—it is simply a collection of nonoverlapping oriented k-cells that fit together in the nicest possible way—the above definition is rather lengthy. This is due to conditions (c), (d), and (e), which are actually redundant—they follow from (a) and (b) and the fact that the k-cells of are oriented (positively with respect to the orientation of M). However, instead of proving this, it will be more convenient for us to include these redundant conditions as hypotheses.
Condition (d) means the following. Suppose that B is the (i, α)th face of Ar, and is also the (j, β)th face of As. If
are orientation-preserving parametrizations of Ar and As respectively, then
What we are assuming in condition (d) is that the mapping
has a negative Jacobian determinant if (− 1)i+α = (− 1)j+β, and a positive one if (−1)i+α = − (−1)j+β. By the proof of Lemma 5.7 , this implies that
for any continuous (k − 1)-form ω. Consequently, if we form the sum
then the integrals over all interior faces cancel in pairs. This conclusion is precisely what will be needed for the proof of Stokes' theorem for an oriented cellulated region.
To see the “visual” significance of condition (d) note that, in Fig. 5.35 , the arrows, indicating the orientations of two adjacent 2-cells, point in opposite directions along their common edge (or face). This condition is not (and cannot be) satisfied by the “cellulation” of the Mobius strip indicated in Fig. 5.36.
Figure 5.35
Figure 5.36
If (R, ) is an oriented cellulated region in the smooth k-manifold , and ω is a continuous differential k-form defined on R, we define the integral of ω on R by
where A1, . . . , Ap are the oriented k-cells of . Note that this might conceivably depend upon the oriented cellulation ; a more complete notation (which, however, we will not use) would be ∫(R, ) ω.
Since the boundary ∂R of R is, by condition (e), the union of all the boundary faces B1, . . . , Bq of the k-cells of , we want to define the integral, of a (k − 1)-form α on ∂R, as a sum of integrals of α over the (k − 1)-cells B1, . . . , Bq. However we must be careful to choose the proper orientation for these boundary faces, as prescribed by the orientation-preserving parametrizations φ1, . . . , φp of the k-cells A1, . . . , Ap of K. If Bs is the (is, s)th face of , then
For brevity, let us write
and
Then is the integral corresponding to the face Bs of , which appears in the sum which defines . We therefore define the integral of the (k − 1)-form α on ∂R by
Although the notation here is quite tedious, the idea is simple enough. We are simply saying that
with the understanding that each boundary face Bs has the orientation that is prescribed for it in the oriented boundary of the oriented k-cell .
This is our final definition of an integral of a differential form. Recall that we originally defined the integral of the k-form α on the k-dimensional surface patch φ by formula (10) in Section 5 . We then defined the integral of α on a k-cell A in an oriented smooth k-manifold M by
where φ : Ik → A is a parametrization that extends to an orientation-preserving coordinate patch for M. The definitions for oriented cellular regions, given above, generalize the definitions for oriented cells, given previously in this section. That is, if (R, ) is an oriented cellular region in which the oriented cellulation consists of the single oriented k-cell R = A, then the integrals
as defined above, reduce to the integrals
as defined previously.
With all this preparation, the proof of Stokes' theorem for oriented cellulated regions is now a triviality.
Theorem 6.3 Let (R, ) be an oriented cellulated region in an oriented smooth k-manifold in n. If α is a differential (k − 1)-form defined on an open set that contains R, then
PROOF Let A1, . . . , Ap be the oriented k-cells of . Then
by Theorem 6.2 , φ1, . . . , φp being the parametrizations of A1, . . . , Ap. But this last sum is equal to
since by Eq. (6) the integrals on interior faces cancel in pairs, while the integral on each boundary face appears once with the “correct” sign—the one given in Eq. (8) .
The most common applications of Stokes' theorem are not to oriented cellulated regions as such, but rather to oriented manifolds-with-boundary. A compact oriented smooth k-manifold-with-boundary is a compact region V in an oriented k-manifold , such that its boundary ∂V is a smooth (compact) (k − 1)-manifold. For example, the unit ball Bn is a compact n-manifold-with-boundary. The ellipsoidal ball
is a compact 3-manifold-with-boundary, as is the solid torus obtained by revolving about the z-axis the-disk in the yz-plane (Fig. 5.37) . The hemisphere is a compact 2-manifold-with-boundary, as is the annulus .
If V is a compact oriented smooth k-manifold-with-boundary contained in the oriented smooth k-manifold , then its boundary ∂V is an orientable (k − 1)-manifold; the positive orientation of ∂V is defined as follows. Given , there exists a coordinate patch φ : U → M such that
(i),
(ii), and
(iii)φ−1(int V) is contained in the open half-space xk > 0 of k.
Figure 5.37
We choose φ to be an orientation-preserving coordinate patch for M if k is even, and an orientation-reversing coordinate patch for M if k is odd. For the existence of such a coordinate patch, see Exercises 4.1 and 4.4 of Chapter III . The reason for the difference between the case k even and the case k odd will appear in the proof of Theorem 6.4 below.
If φ is such a coordinate patch for M, then is a coordinate patch for the (k − 1)-manifold ∂V (see Fig. 5.38) . If φ1, . . . , φm is a collection of such coordinate patches for M, whose images cover ∂V, then their restrictions φ1, . . . , φm to k−1 form an atlas for ∂V. Since the fact that φi and φj overlap positively (because either both are orientation-preserving or both are orientation-reversing) implies that φi and φj overlap positively, this atlas {φi} is an orientation for ∂V. It is (by definition) the positive orientation of ∂V.
Figure 5.38
As a useful exercise, the reader should check that this construction yields the counterclockwise orientation for the unit circle S1, considered as the boundary of the disk B2 in the oriented 2-manifold 2, and, more generally, that it yields the positive orientation (of Section 2) for any compact 2-manifold-with-boundary in 2.
A cellulation is a special kind of paving, and we have previously stated that every compact smooth manifold has a paving. It is, in fact, true that every oriented compact smooth manifold-with-boundary possesses an oriented cellulation. If we accept without proof this difficult and deep theorem, we can establish Stokes' theorem for manifolds-with-boundary.
Theorem 6.4 Let V be an oriented compact smooth k-manifold-with-boundary in the oriented smooth k-manifold . If ∂V has the positive orientation, and α is a differential (k − 1)-form on an open set containing V, then
PROOF Let = {A1, . . . , Ap} be an oriented cellulation of V. We may assume without loss of generality that the (k, 0)th faces of the first q of these oriented k-cells are the boundary faces of the cellulation. Let φi: Ik → Ai be the given orientation-preserving parametrization of Ai. Then
by Theorem 6.3 .
By the definition in Section 5 , of the integral of a differential form over an oriented manifold,
where B1, . . . , Bs are the oriented (k − 1)-cells of any oriented paving of the oriented (k − 1)-manifold ∂V. That is,
if i is an orientation-preserving parametrization of Bi, i = 1, . . . , s.
Now the boundary faces constitute a paving of ∂V; the question here is whether or not their parametrizations are orientation-preserving. But from the definition of the positive orientation for ∂V, we see that is orientation-preserving or orientation-reversing according as k is even or odd respectively, since φi : Ik → M is orientation-preserving in either case (Fig. 5.39) . Therefore
by Eq. (11) .
Figure 5.39
There is an alternative proof of Theorem 6.4 which does not make use of oriented cellulations. Instead of splitting the manifold-with-boundary V into oriented cells, this alternative proof employs the device of “partitions of unity” to split the given differential form α into a sum
where each of the differential forms αi vanishes outside of some k-cell. Although the partitions of unity approach is not so conceptual as the oriented cellulations approach, it is theoretically preferable because the proof of the existence of partitions of unity is much easier than the proof of the existence of oriented cellulations. For the partitions of unity approach to Stokes' theorem, we refer the reader to Spivak, “Calculus on Manifolds” [18].
Example Let D be a compact smooth n-manifold-with-boundary in n. If
then
Theorem 6.4 therefore gives
so
if ∂D is positively oriented. Formula (12) is the n-dimensional generalization of the formula
for the area of a region (see Example 2 of Section 2) .
According to Exercise 5.13 , the differential (n − 1)-form
is the surface area form of the unit sphere . With D = Bn, formula (12) therefore gives
the relationship that we have previously seen in our explicit (and separate) computations of v(Bn) and a(Sn−1).
The applications of Stokes' theorem are numerous, diverse, and significant. A number of them will be treated in the following exercises, and in subsequent sections.
Exercises
6.1Let V be a compact oriented smooth (k + l + 1)-dimensional manifold-with-boundary in n. If α is a k-form and β an l-form, each in a neighborhood of V, use Stokes' theorem to prove the “integration by parts” formula
6.2If ω is a differential k-form on n such that
for every compact oriented smooth k-manifold , use Stokes' theorem to show that ω is closed, that is, dω = 0.
6.3Let α be a differential (k − 1)-form defined in a neighborhood of the oriented compact smooth k-manifold . Then prove that
Hint: If B is a smooth k-dimensional ball in M (see Fig. 5.40) , and V = M − int B, then
Figure 5.40
6.4Let V1 and V2 be two compact n-manifolds-with-boundary in n such that . If α is a differential (n − 1)-form such that dα = 0 on W = V1 − int V2, show that
if ∂V1 and ∂V2 are both positively oriented.
6.5We consider in this exercise the differential (n − 1)-form dθ defined on n − 0 by
where ρ2 = x12 + x22 + · · · + xn2. For reasons that will appear in the following exercise, this differential form is called the solid angle form on n, and this is the reason for the notation dθ. Note that dθ reduces to the familiar
in the case n = 2.
(a)Show that dθ is closed, d(dθ) = 0.
(b)Note that, on the unit sphere Sn−1, dθ equals the surface area form of Sn−1. Conclude from Exercise 6.3 that dθ is not exact, that is, there does not exist a differential (n − 2)-form α on n − 0 such that dα = dθ.
(c)If M is an oriented compact smooth (n − 1)-manifold in n which does not enclose the origin, show that
(d)If M is an oriented compact smooth (n − 1)-manifold in n which does enclose the origin, use Exercise 6.4 to show that
6.6Let be an oriented compact (n − 1)-dimensional manifold-with-boundary such that every ray through 0 intersects M in at most one point. The union of those rays through 0 which do intersect M is a “solid cone” C(M). We assume that the intersection , of C(M) and the sphere Sρn−1 of radius ρ (Fig. 5.41) , is an oriented compact (n − 1)-manifold-with-boundary in Sρn−1 (with its positive orientation). The solid angle θ (M) subtended by M is defined by
(a)Show that θ(M) = 1/ρn−1a(Nρ).
(b)Prove that θ(M) = ∫M dθ.
Hints: Choose ρ > 0 sufficiently small that Nρ is “nearer to 0” than M. Let V be the compact region in n that is bounded by , where R denotes the portion of the boundary of C(M) which lies between M and Nρ. Assuming that V has an oriented cellulation, apply Stokes' theorem. To show that
let {A1, . . . , Am} be an oriented paving of ∂Nρ. Then {B1, . . . , Bn) is an oriented paving of R, where . Denote by l(x) the length of the segment between Nρ and M, of the ray through . If
is a parametrization of Ai, and
is defined by φi(x, t) = φi(x), finally show that
Figure 5.41
6.7If V is a compact smooth n-dimensional manifold-with-boundary in n, then there are two unit normal vector fields on ∂V—the outer normal, which at each point of ∂V points out of V, and the inner normal, which points into V. On the other hand, if M is an oriented smooth (n − 1)-manifold in n, then we have seen in Exercise 5.13 that the orientation of M uniquely determines a normal vector field N on M. If M = ∂V is positively oriented (as the boundary of V), show that N is the outer normal vector field. Hence we can conclude from Exercise 5.13 that the surface area form of ∂V is
where the ni are the components of the outer normal vector field on ∂V.